Spectral approximation for the separable covariance mixture model

TL;DR

This paper develops a spectral approximation method for the separable covariance mixture model, providing non-asymptotic results without requiring diagonalizability, and describes the limiting spectral distribution of related sample covariance matrices.

Contribution

It introduces a novel spectral approximation framework for the separable covariance mixture model that relaxes previous diagonalizability assumptions and includes asymptotic spectral distribution results.

Findings

Resolvent matrices approximate deterministic equivalents under certain conditions.

Non-asymptotic results hold without simultaneous diagonalizability.

Asymptotic spectral distribution of sample covariance matrices is characterized.

Abstract

This paper introduces the separable covariance mixture model, which assumes a data-matrix $Y$ to be of the form $$ \sum\limits_{r=1}^R A_r X B_r $$ for one random $(d \times n)$-matrix $X$ with independent centered variance-one entries, and for two families of deterministic matrices $A_1,\dots,A_R \in \mathbb{C}^{d \times d}$ and $B_1,\dots,B_R \in \mathbb{C}^{n \times n}$. Under certain assumptions, it is shown that the resolvents $(\frac{1}{n} Y Y^* - z \operatorname{Id}_d)^{-1}$ and $(\frac{1}{n} Y^* Y - z \operatorname{Id}_n)^{-1}$ respectively approximate the deterministic matrices $$ -\frac{1}{z}\Big( \operatorname{Id}_d + \sum\limits_{r,s=1}^R \delta^{(B)}_{r,s}(z) A_{r} A_{s}^* \Big)^{-1} \ \ \text{ and } \ \ -\frac{1}{z}\Big( \operatorname{Id}_n + \sum\limits_{r,s=1}^R \delta^{(A)}_{r,s}(z) B_{s}^*B_{r} \Big)^{-1} \ , $$ where $\delta^{(A)}, \delta^{(B)} \in \mathbb{C}^{R \times R}$ are uniquely defined solutions to a certain dual system of equations. The results are non-asymptotic and do not require simultaneous diagonalizability of the families $(A_r)_{r \leq R}$ or $(B_r)_{r \leq R}$, as was required in previous works such as [Hazarika and Paul (2025)] or [Mei et al. (2023)]. An asymptotic application, which describes the limiting spectral distribution of the sample covariance matrix analogues $\frac{1}{n} Y Y^*$ or $\frac{1}{n} Y^* Y$, is included.